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1. Consider the monopolist from the last problem set Y ho is serving tY o types of consumers Y ith demand functions.

Let c(y) = 0:

Recall from the last problem set that an agents utility from consuming good 1 is given by:

(a) pH , kL,and kH .can all be different Dray the optimal price and shade the producer surpluses. Will the agent serve both markets? Suppose that the loY market all have student IDs that alloY them to be differentiated from the high types. For an agent to participate it must be that:

These are knoY n collectively as individual rationality (IR) constraints or participation graphically the folloY ing cases. You do not have to do the math for this problem:

constraints (P C). Dray

i. Suppose legislation requires that k =0 and that pL = pH . Dray the optimal price and shade the producer surplus. ii. Suppose legislation requires that k = 0 but pL and pH can be different. Dray the optimal prices and shade the producer sur- pluses iii.Suppose legislation requires pL = pH and k L = kH . Dray the optimal price and shade the producer surpluses. Will the agent serve both markets? iv. Suppose both pL,

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(b) NoY suppose that the agent can not tell one type of consumer from the other. In this case ye require that each agent type gets more utility from choosing the bundle meant for them than the one meant for the other type of agent:

UL (pL , kL ) ≥ UL (pH , kH ) UH (pH , kH ) ≥ UH (pL , kL ) These conditions are called incentive compatability constraints. They bascially say that a high type y ho pretends to be a loY type must get less utility than he y ould acting as a high type. In standard screening problems it is typically the IR constraint of the loY type and the IC constraint of the high type that binds. That is:

UL (p, k) ≥ 0 UH (pH , kH ) ≥ UH (pL , kL ) To think about these concepts dray graphically the folloY ing cases. You do not have to do math for this problem. If you haven't already done it, assume that the loY market is large enough that it isn't shut out.: i. Suppose legislation requires pL = pH and k L = kH . Dray the optimal price and shade the producer surpluses. Will the agent serve both markets? Is this different from the problem Yith full signals above? ii. Suppose both pL, pH , kL,and kH .can all be different Dray the optimal price and shade the producer surpluses. iii.Suppose instead of only being able to restrict p and k the mo- nopolist can also restrict quantity qL and qH . Dray the optimal price and shade the producer surpluses.

2.××(From McAfee 1996) In May 1990, IBM

announced the introduction of the LaserPrinter E, a loYer cost alternative to its popular LaserPrinter. The LaserPrinter E Yas identical to the original LaserPrinter except that it printed at 5 ppm instead of 10 ppm. According to Jones(1990), the engine and parts of the printer Yere virtually identical to the faster printer except that the controller for the sloYer printer had firmYear that inserted Yait states to sloY the print speed of the printer. This problem is designed to see hoY such damaging of a product may make everyone better off. (a) Assume first that you have a single high quality product that you are selling. The product has constant marginal cost to produce of 81. There are tYo types of consumers Yho are yilling to buy a single unit of your good. 25% of the population is type 1 and are yilling to buy your product if the cost of producing it is less than or equal to 811. 75% of the population is type 2 and are yilling to buy your product

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all s> 0 (it is costly to produce an inferior goif it is priced at 83.00. Suppose that you must sell your product at a constant price (you can not screen your consumers) - shoY that you Y ill only sell to the high types. (b) NoY assume that you can produce a broken version of your product. Let s be a measure of hoY broken your product is. The high types get utility UH (xi , s) for consuming one unit of type s good. Thus UH (1, 0) is the value to the high type of buying one unit of the original quality good and UH (1, s) Y ould be the utility of the high type for buying a good Y ith quality s. Similarly, the loY type get UL (1, 0) for the original good and UL (1, s) for consuming a loY good od). i. Again assume that there is no screening other than offering a loY quality good. Set up the monopolists problem. ii. Suppose that UH (1, s) = 11 — 2s, UL (1, s) = 3 — .25s, and c(s) = 1. Find the optimal quality of the loYer good and the amount charged for the tYo goods. iii.Suppose by lay the monopolist can only reduce quality doYn to s = 3. ShoY that in this case, everyone is at least as Yell off due to the creation of the damaged good.

2. Suppose that a hospital carries K doses of an opium based medicine used to help patients suffering from micro fibralgia. N ≥ K patients come seeking the drug at the same time. 6 of these actually suffer from the disease yhile N-6 of them are drug addicts looking for a fix. (a) Suppose that the hospital does not care about the utility of the drug addicts. The utility of a micro fibralgia agent receiving the drug is 10. The utility of a micro fibralgia agent Yho does not receive the drug is zero. Calculate the total expected utility the hospital can provide to its micro fibralgia patients if it has no Yay to screen betYeen drug addicts and true patients. (b) Suppose that the Hospital can force agents to Yaste time before being treated. The utility of a micro fibralgia user Yho must Yait and receives the medication yith probability p is:

UM icr o(r, p) = 10p — 2r Drug addicts have a utility function of:

UD A (r, p) = 4 — r The addicts are not rational - as long as they stay in the hospital they believe that the probability of receiving treatment is 1.

Both agents outside option is zero, that ismatlabassignmentexperts.com UMicro(0, 0) = UDA(0, 0) = 0.

i. Suppose the hospitals objective function is to maximize the util- ity of its micro fibralgia patients. Argue that the only possible optimal Yait time they could choose to impose is Y=0,4. (5 points) ii. Calculate the Expected Utility of the micro fibralgia patients and drug addicts at each of the yait times as a function of N and K. (5 points) iii.Suppose that K = 10, N=60 - should the hospital screen its patients? Will there be a shortage of medication? (5 points) 4.

×× Consider

a y orld in y hich state contingent contracts can be yritten. There are tY o people (Chris and Tatiana) and tY o goods - umbrellas and sY im suits. When it is raining, both people like having as many um- brellas as possible.

If it is sunny, both people like having as many sY im suits as possible. Suppose each person has an identical utility function U(u, s�r�i���) i = u, U(u, s�su��y) = s Y here u and s are the number of umbrellas and sY imming suits the agent possesses in the given state (they get more utility from having more umbrellas).

Assume that at t=0 each agent is endoY ed Y ith 1 umbrella and 1 sY im- suit for the next period. The tY o agents disagree on the probability of Y hether it Y ill rain in the future. Tatiana is an optimist and believes it is going to be sunny Y ith probability .75 and rainy Y ith probability .25. Chris is a pessamist and believes it is going to be sunny Y ith probability .5 and rain yith probability .5. Both people are vNM utility maximizers, thus:

(a)Dray the edgeyorth box for this problem. What do indifference curves for the tYo individuals look like? (b)Before doing any math - Yhy is it that Ye expect our final allocation to be on the boundary? (c)Why in this problem do Ye not have a unique price on yhich all markets clear? (d)Suppose that I am a social planner trying to efficiently allocate um- brellas and bathing suits to my housemates. I solve: max

λ(UT atiana (ut, s t ))+ (1 — λ)(UChris(uc, sc))

ShoY that all the pareto optimal outcomes are on the boundary of the edgeyorth box.

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5. Suppose that Ye are in an economy Yith tYo individuals i = 1, 2 each Yith identical utility function ui (x1 , x2 ) = (x1 x2 )1/2 . (a) Suppose that agent 1 is endoYed Yith x1 = 1, x2 = 0 and agent 2 isendoY ed Y ith x1 = 0, x2 = 2. At Y hat price ratio p1

Y ill prices clear in this market? Determine the allocation in this situation.

(b) Prove that in this economy, the price found in part (a) Yill clear the market regardless of the initial allocation as long as the total endoYment stays the same. What Yill the contract curve look like? (c) ShoY that the contract curve and the price function are orthoginal in this problem.

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P

1) Part a (i).

3rd Degree Price Discrimination PL  P H , K L  K H  0 * Inefficiency in both markets *Price between the monopoly price w/ Seperate Markets q(L)

q(H)

Deadweight Loss in Low Market

Q Deadweight Loss in Low Market pq

T 2V (q)  T

1V (q)  T

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q

P 1) Part a (ii).

3rd Degree Price Discrimination P L  PH , K L  K H  0 * Inefficiency in both markets * Priced as independent Monopolies q(L)

q(H)

Deadweight Loss in Low Market

Q Deadweight Loss in Low Market pq

T 2V (q)  T 1V (q)  T

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q

P Fixed Fee

1) Part a (iii).

3rd Degree Price Discrimination

q(L)

q(H)

Deadweight Loss in Low Market

PL  PH , K L  K H * Inefficiency in both markets *Price Decreases from part (i) due to value of fixed fees Q Deadweight Loss in Low Market pq

T 2V (q)  T

1V (q)  T

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q

P Fixed Fee

1) Part a (iii).

1st Degree Price Discrimination PL  PH , K L  K H * Efficiency in both markets * Price = Marginal Cost (zero)

q(L)

q(H)

Q

pq

T 2V (q)  T

1V (q)  T

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q

P Fixed Fee

1) Part b (i).

2rd Degree Price Discrimination

q(L)

q(H)

Deadweight Loss in Low Market

P L  PH , K L  K H * Inefficiency in both markets *Note that this is identical to part a (iii) Q Deadweight Loss in Low Market pq

T 2V (q)  T

1V (q)  T

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q

P Low Fixed High Fixed

q(L)

q(H)

1) Part b (ii).

2rd Degree Price Discrimination PL  P H , K L  K H * Inefficiency in low market * Efficiency in high market *IC constraint gives High market Positive Rents Q

Deadweight Loss in Low Market pq

T 2V (q)  T

1V (q)  T

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q

P

1) Part b (iii).

Low Fixed

2rd Degree Price Discrimination High Fixed

q(L)

q(H)

PL  PH , K L  K H , QL constrained * Inefficiency in low market * Efficiency in high market *IC constraint gives High market Positive Rents monopolists Rents * Quality constraints increases

Deadweight Loss in Low Market pq

T 2V (q)  T

1V (q)  T

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q

1.Solutions 2. (a) If the monopolist serves both markets, it Yould solve:

in this case, r h = rl = 3.00 and his profits Y ould be π(rh = rl = 3) = 82 If the monopolist serves only the high market it solves:

and Y ould thus set r h = 11. In this case, π(rh = rl = 11) = .25 × 10 = 2.50.Thus the monopolist Y ill serve only the high market. i. The monopolist solves:

Notice that if U l (1, sl ) = rl and Uh (1, sh ) — rh = Uh (1, sl ) — rl → Uh (1, sh ) > rh thus the IR constraint for the high type y on't bind. Substitution of IC H and IR L into IC L yields: 0 ≥ Ul (1, sh ) — Uh (1, sh ) + Uh (1, sl ) — Ul (1, sl ) Rearranging ye have: Uh (1, sh ) — Uh (1, sl ) ≥ Ul (1, sh ) — Ul (1, sl ) Dividing both sides by sh — sl (a negative number) Y e have: Uh (1, sh ) matlabassignmentexperts.com — Uh (1, sl ) Ul (1, sh ) — Ul (1, sl ) sh — sl

sh — sl

Substituion of the utility functions in (1) and (2) gives us:

Substitution of (2) into (1) yields:

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ii. Define U(r, p) as the utility an agent gets Y ith a y ait time r imposed and a probability of receiving a drug p based on that Y ait time. Suppose that r = 4. We have tY o cases. Suppose that K > 6. When the agent screens, every patient gets a dose. T hus : E (U(r, 1)) = 10 — 4r = 2 Suppose that K < 6. Then K E (U(r, 6

K K ) = 10 — 4r = 10 — 8 6 6

We can yrite these tY o cases as: K E (U(4, min( 6

K , 1)) = 10 min( , 1) — 8 6

When y = 0 all the agents stay. Thus

Chris's better than set is to the North West. Tatiana better than set is to the South East. Thus the area to the north east Y ould make the both better off than autarky (b) Consider any other point inside the region in yhich both people yould trade that is not on the boundary. Notice that there is alYays a point to the north Yest Yill be better for both participants. Thus the pareto optimal set Yill be on the boundary. (c) Uniqueness of a price is guaranteed Yhen the pareto optimal point is on the interior of the Edgeyorth box. In this problem, all outcomes are boundary condiditons and thus there are a lot of price vectors that Yill lead to a situation yhere markets clear. (d) The social planner maximizes:

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